NAME:Math 103 Exam 4 retake12 December 2008100 pointsInstructions:1. This exam has 7 pages (including this one and the formula sheet), which contain 7 questionsand 2 bonus questions. Please check that you have all of the pages.2. Answer all of the following questions clearly and completely. Justify all of your answers.3. You may not use a book or any notes for this exam, except the formula sheet attached as thelast page of the exam.4. Give your answer to each problem completely and clearly in the space provided. You may usethe back of the exam pages for scratch work; however, if you want this work to be considered,make note of it in the space provided for the problem.5. Erase or cross out work you do not wish to be graded.6. Credit, partial or full, will be given only if sufficient steps leading to the answers are shown.7. You have 50 minutes to complete this exam.Page 1aAbBcCQuestion 1. (20 points) Solve the following triangles.(a) (10 points) b = 4, c = 5, B = 95◦(b) (10 points) a = 2, b = 2, C = 50◦Page 2Question 2. (20 points) Find the exact values of the following expressions.(a) (5 points) csc7π4(b) (5 points) cos−1 −√32!(c) (5 points) tan−1tan16π9(d) (5 points) tan3π8Page 3Question 3. (20 points) Consider the equation 4 sin2θ + 1 = 4.(a) (10 points) Find the general solution to this equation (with no restriction on the value of θ).(b) (10 points) Find the solutions to this equation on the interval 0 ≤ θ < 2π.Question 4. (10 points) Establish the identity: csc(2θ) =12sec θ csc θ.Page 4Question 5. (10 points) If tan θ = −1880and θ is in Quadrant II, then what are sin θ and cos θ?Question 6. (10 points) Let f(x) = −4 sin(5πx). What are the amplitude and the period of f ?Page 5Question 7. (10 points) In the figure below, the center of the circle is one of the corners of thetriangle. Find the area of the shaded triangle.7 in.4 in.5 in.Bonus. (+3 points) Explain whycos 1◦+ cos 2◦+ cos 3◦+ ··· + cos 358◦+ cos 359◦= −1.Bonus. (+1 point) Find a common English word containing the letters KSG together and in thatorder.Page 6Sum and difference formulassin(α + β) = sin α cos β + cos α sin βsin(α − β) = sin α cos β − cos α sin βcos(α + β) = cos α cos β − sin α sin βcos(α − β) = cos α cos β + sin α sin βtan(α + β) =tan α + tan β1 − tan α tan βtan(α − β) =tan α − tan β1 + tan α tan βDouble-angle formulassin(2θ) = 2 sin θ cos θcos(2θ) = cos2θ − sin2θcos(2θ) = 1 − 2 sin2θcos(2θ) = 2 cos2θ − 1tan(2θ) =2 tan θ1 − tan2θCorollaries of double-angle formulassin2θ =1 − cos(2θ)2cos2θ =1 + cos(2θ)2tan2θ =1 − cos(2θ)1 + cos(2θ)Half-angle formulassinα2= ±r1 − cos α2cosα2= ±r1 + cos α2tanα2= ±r1 − cos α1 + cos αtanα2=1 − cos αsin α=sin α1 + cos αProduct-to-sum formulassin α sin β =12cos(α − β) − cos(α + β) cos α cos β =12cos(α − β) + cos(α + β) sin α cos β =12sin(α + β) + sin(α − β) Sum-to-product formulassin α + sin β = 2 sinα + β2cosα − β2sin α − sin β = 2 sinα − β2cosα + β2cos α + cos β = 2 cosα + β2cosα − β2cos α − cos β = −2 sinα + β2sinα − β2Law of Sinessin Aa=sin Bb=sin CcLaw of Cosinesc2= a2+ b2− 2ab cos Cb2= a2+ c2− 2ac cos Ba2= b2+ c2− 2bc cos AArea of a triangleIn the following formulas, K denotes thearea of a triangle.K =12bhK =12ab sin CK =12bc sin AK =12ac sin BK =ps(s − a)(s − b)(s − c),where s =a + b +